A304465 -id:A304465 - cp's OEIS frontend

A304634 Numbers n with prime omicron 2, meaning A304465(n) = 2.

4, 6, 9, 10, 12, 14, 15, 18, 20, 21, 22, 24, 25, 26, 28, 33, 34, 35, 36, 38, 39, 40, 44, 45, 46, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 68, 69, 72, 74, 75, 76, 77, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 104

Offset: 1

Gus Wiseman, May 15 2018

nonn

If n > 1 is not a prime number, we have A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= omicron(n) > 1 where Omega = A001222, omega = A001221, and omicron = A304465.

			This is a list of normalized factorizations (see A112798) of selected entries:
    4: {1,1}
    6: {1,2}
   12: {1,1,2}
   24: {1,1,1,2}
   36: {1,1,2,2}
   48: {1,1,1,1,2}
   60: {1,1,2,3}
   72: {1,1,1,2,2}
   96: {1,1,1,1,1,2}
  120: {1,1,1,2,3}
  144: {1,1,1,1,2,2}
  180: {1,1,2,2,3}
  192: {1,1,1,1,1,1,2}
  216: {1,1,1,2,2,2}
  240: {1,1,1,1,2,3}
  288: {1,1,1,1,1,2,2}
  384: {1,1,1,1,1,1,1,2}
  420: {1,1,2,3,4}
  432: {1,1,1,1,2,2,2}
  480: {1,1,1,1,1,2,3}
  576: {1,1,1,1,1,1,2,2}
  768: {1,1,1,1,1,1,1,1,2}
  840: {1,1,1,2,3,4}
  864: {1,1,1,1,1,2,2,2}
  960: {1,1,1,1,1,1,2,3}

Cf. A001221, A001222, A001358, A005117, A007774, A007916, A055932, A071625, A112798, A181819, A182850, A182857, A275870, A304464, A304465, A304636, A304647.

Join@@Position[Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,100}],2]

A304636 Numbers n with prime omicron 3, meaning A304465(n) = 3.

Original entry on oeis.org

8, 27, 30, 42, 66, 70, 78, 102, 105, 110, 114, 125, 130, 138, 154, 165, 170, 174, 182, 186, 190, 195, 222, 230, 231, 238, 246, 255, 258, 266, 273, 282, 285, 286, 290, 310, 318, 322, 343, 345, 354, 357, 360, 366, 370, 374, 385, 399, 402, 406, 410, 418, 426

Offset: 1

Gus Wiseman, May 15 2018

nonn

If n > 1 is not a prime number, we have A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= omicron(n) > 1 where Omega = A001222, omega = A001221, and omicron = A304465.

			This is a list of normalized factorizations (see A112798) of selected entries:
     8: {1,1,1}
    30: {1,2,3}
   360: {1,1,1,2,2,3}
   720: {1,1,1,1,2,2,3}
   900: {1,1,2,2,3,3}
  1440: {1,1,1,1,1,2,2,3}
  2160: {1,1,1,1,2,2,2,3}
  2880: {1,1,1,1,1,1,2,2,3}
  4320: {1,1,1,1,1,2,2,2,3}
  5760: {1,1,1,1,1,1,1,2,2,3}
  8640: {1,1,1,1,1,1,2,2,2,3}
Starting with A112798(1801800) and repeatedly taking the multiset of multiplicities we have {1,1,1,2,2,3,3,4,5,6} -> {1,1,1,2,2,3} -> {1,2,3} -> {1,1,1} -> {3}, so 1801800 belongs to the sequence.

Cf. A001221, A001222, A005117, A007916, A014612, A033992, A071625, A112798, A181819, A182850, A182857, A304464, A304465, A304634, A304647.

Join@@Position[Table[Switch[n,1,0,?PrimeQ,1,,NestWhile[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]],Length[#]>1&]//First],{n,200}],3]

A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 5, 3, 4, 2, 6, 2, 4, 4, 7, 2, 6, 2, 6, 4, 4, 2, 10, 3, 4, 5, 6, 2, 8, 2, 11, 4, 4, 4, 9, 2, 4, 4, 10, 2, 8, 2, 6, 6, 4, 2, 14, 3, 6, 4, 6, 2, 10, 4, 10, 4, 4, 2, 12, 2, 4, 6, 13, 4, 8, 2, 6, 4, 8, 2, 15, 2, 4, 6, 6, 4, 8, 2, 14, 7, 4, 2, 12, 4, 4, 4, 10, 2, 12, 4, 6, 4, 4, 4, 22, 2, 6, 6, 9, 2, 8, 2, 10, 8

Offset: 1

Matthew Vandermast, Dec 07 2010

a(n) depends only on prime signature of n (cf. A025487). a(m) = a(n) iff m and n have the same prime signature, i.e., iff A046523(m) = A046523(n).

Because A046523 (the smallest representative of prime signature of n) and this sequence are functions of each other as A046523(n) = A181821(a(n)) and a(n) = a(A046523(n)), it implies that for all i, j: a(i) = a(j) <=> A046523(i) = A046523(j) <=> A101296(i) = A101296(j), i.e., that equivalence-class-wise this is equal to A101296, and furthermore, applying any function f on this sequence gives us a sequence b(n) = f(a(n)) whose equivalence class partitioning is equal to or coarser than that of A101296, i.e., b is then a sequence that depends only on the prime signature of n (the multiset of exponents of its prime factors), although not necessarily in a very intuitive way. - Antti Karttunen, Apr 28 2022

			20 = 2^2*5 has the exponents (2,1) in its prime factorization. Accordingly, a(20) = prime(2)*prime(1) = A000040(2)*A000040(1) = 3*2 = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n

Column 1 of A353510 (see also A325239 and A325277).
Cf. A000040, A000265, A001511, A001222, A003963, A005361, A007814, A008578, A028234, A046523, A056239, A064553, A064989, A067029, A101296 (restricted growth sequence transform), A108951, A122111, A124010, A124859, A156552, A181820, A181821, A182850, A182855, A182857 (also A323014), A115621, A101296, A238690, A238745, A238747, A238748, A246029, A304465, A304647, A305732, A305733, A320118, A323022, A325501, A325502, A325507, A325508, A325755 (A353566), A325756, A328830 [= a(a(n))], A328835, A351564 (characteristic function of A130091), A351944, A351946, A353379.
Left inverse of A304660.

a181819 = product . map a000040 . a124010_row
-- Reinhard Zumkeller, Mar 26 2012

A181819 := proc(n)
    local a;
    a := 1;
    for pf in ifactors(n)[2] do
        a := a*ithprime(pf[2]) ;
    end do:
    a ;
end proc:
seq(A181819(n),n=1..80) ; # R. J. Mathar, Jan 09 2019
# second Maple program:
a:= n-> mul(ithprime(i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..105);  # Alois P. Heinz, Nov 26 2024

{1}~Join~Table[Times @@ Prime@ Map[Last, FactorInteger@ n], {n, 2, 120}] (* Michael De Vlieger, Feb 07 2016 *)

a(n) = {my(f=factor(n)); prod(k=1, #f~, prime(f[k,2]));} \\ Michel Marcus, Nov 16 2015

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) (else (* (A000040 (A067029 n)) (A181819 (A028234 n))))))
;; Antti Karttunen, Feb 05 2016

;; With memoization-macro definec.
(definec (A181819 n) (cond ((= 1 n) 1) ((even? n) (* (A000040 (A007814 n)) (A181819 (A000265 n)))) (else (A181819 (A064989 n)))))
;; Antti Karttunen, Feb 07 2016

From Antti Karttunen, Feb 07 2016: (Start)
a(1) = 1; for n > 1, a(n) = A000040(A067029(n)) * a(A028234(n)).
a(1) = 1; for n > 1, a(n) = A008578(A001511(n)) * a(A064989(n)).
Other identities. For all n >= 1:
a(A124859(n)) = A122111(a(n)) = A238745(n). - from Matthew Vandermast's formulas for the latter sequence.
(End)
a(n) = A246029(A156552(n)). - Antti Karttunen, Oct 15 2016
From Antti Karttunen, Apr 28 & Apr 30 2022: (Start)
A181821(a(n)) = A046523(n) and a(A046523(n)) = a(n). [See comments]
a(n) = A329900(A124859(n)) = A319626(A124859(n)).
a(n) = A246029(A156552(n)).
a(a(n)) = A328830(n).
a(A304660(n)) = n.
a(A108951(n)) = A122111(n).
a(A185633(n)) = A322312(n).
a(A025487(n)) = A181820(n).
a(A276076(n)) = A275735(n) and a(A276086(n)) = A328835(n).
As the sequence converts prime exponents to prime indices, it effects the following mappings:
A001221(a(n)) = A071625(n). [Number of distinct indices --> Number of distinct exponents]
A001222(a(n)) = A001221(n). [Number of indices (i.e., the number of prime factors with multiplicity) --> Number of exponents (i.e., the number of distinct prime factors)]
A056239(a(n)) = A001222(n). [Sum of indices --> Sum of exponents]
A066328(a(n)) = A136565(n). [Sum of distinct indices --> Sum of distinct exponents]
A003963(a(n)) = A005361(n). [Product of indices --> Product of exponents]
A290103(a(n)) = A072411(n). [LCM of indices --> LCM of exponents]
A156061(a(n)) = A290107(n). [Product of distinct indices --> Product of distinct exponents]
A257993(a(n)) = A134193(n). [Index of the least prime not dividing n --> The least number not among the exponents]
A055396(a(n)) = A051904(n). [Index of the least prime dividing n --> Minimal exponent]
A061395(a(n)) = A051903(n). [Index of the greatest prime dividing n --> Maximal exponent]
A008966(a(n)) = A351564(n). [All indices are distinct (i.e., n is squarefree) --> All exponents are distinct]
A007814(a(n)) = A056169(n). [Number of occurrences of index 1 (i.e., the 2-adic valuation of n) --> Number of occurrences of exponent 1]
A056169(a(n)) = A136567(n). [Number of unitary prime divisors --> Number of exponents occurring only once]
A064989(a(n)) = a(A003557(n)) = A295879(n). [Indices decremented after <--> Exponents decremented before]
Other mappings:
A007947(a(n)) = a(A328400(n)) = A329601(n).
A181821(A007947(a(n))) = A328400(n).
A064553(a(n)) = A000005(n) and A000005(a(n)) = A182860(n).
A051903(a(n)) = A351946(n).
A003557(a(n)) = A351944(n).
A258851(a(n)) = A353379(n).
A008480(a(n)) = A309004(n).
a(A325501(n)) = A325507(n) and a(A325502(n)) = A038754(n+1).
a(n!) = A325508(n).
(End)

Name "Prime shadow" (coined by Gus Wiseman in A325755) prefixed to the definition by Antti Karttunen, Apr 27 2022

A182850 a(n) = number of iterations that n requires to reach a fixed point under the x -> A181819(x) map.

Original entry on oeis.org

0, 0, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 2, 4, 1, 3, 1, 2, 3, 3, 3, 3, 1, 3, 3, 4, 1, 3, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 5, 1, 3, 4, 2, 3, 3, 1, 4, 3, 3, 1, 4, 1, 3, 4, 4, 3, 3, 1, 4, 2, 3, 1, 5, 3, 3, 3, 4, 1, 5, 3, 4, 3, 3, 3, 4, 1, 4, 4, 3, 1, 3, 1, 4, 3

Offset: 1

Matthew Vandermast, Jan 04 2011

nonn

The fixed points of the x -> A181819(x) map are 1 and 2. Note that the x -> A000005(x) map has the same fixed points, and that A000005(n) = A181819(n) iff n is cubefree (cf. A004709). Under the x -> A181819(x) map, it seems significantly easier to generalize about which kinds of integers take a given number of iterations to reach a fixed point than under the x -> A000005(x) map.

Also the number of steps in the reduction of the multiset of prime factors of n wherein one repeatedly takes the multiset of multiplicities. For example, the a(90) = 5 steps are {2,3,3,5} -> {1,1,2} -> {1,2} -> {1,1} -> {2} -> {1}. - Gus Wiseman, May 13 2018

			A181819(6) = 4; A181819(4) = 3; A181819(3) = 2; A181819(2) = 2. Therefore, a(6) = 3, a(4) = 2, a(3)= 1, and a(2) = 0.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Fixed Point
Eric Weisstein's World of Mathematics, Map

A182857 gives values of n where a(n) increases to a record.

Cf. A000961, A001222, A003434, A005117, A007916, A036459, A112798, A130091, A181819, A182851-A182858, A238748, A304455, A304464, A304465.

a182850 n = length $ takeWhile (`notElem` [1,2]) $ iterate a181819 n
-- Reinhard Zumkeller, Mar 26 2012

Table[If[n<=2,0,Length[FixedPointList[Sort[Length/@Split[#]]&,Sort[Last/@FactorInteger[n]]]]-1],{n,100}] (* Gus Wiseman, May 13 2018 *)

;; With memoization-macro definec.
(definec (A182850 n) (if (<= n 2) 0 (+ 1 (A182850 (A181819 n))))) ;; Antti Karttunen, Feb 05 2016

For n > 2, a(n) = a(A181819(n)) + 1.
a(n) = 0 iff n equals 1 or 2.
a(n) = 1 iff n is an odd prime (A000040(n) for n>1).
a(n) = 2 iff n is a composite perfect prime power (A025475(n) for n>1).
a(n) = 3 iff n is a squarefree composite integer or a power of a squarefree composite integer (cf. A182853).
a(n) = 4 iff n's prime signature a) contains at least two distinct numbers, and b) contains no number that occurs less often than any other number (cf. A182854).

A323014 a(1) = 0; a(prime) = 1; otherwise a(n) = 1 + a(A181819(n)).

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 2, 4, 1, 3, 1, 2, 3, 3, 3, 3, 1, 3, 3, 4, 1, 3, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 5, 1, 3, 4, 2, 3, 3, 1, 4, 3, 3, 1, 4, 1, 3, 4, 4, 3, 3, 1, 4, 2, 3, 1, 5, 3, 3, 3, 4, 1, 5, 3, 4, 3, 3, 3, 4, 1, 4, 4, 3, 1, 3, 1, 4, 3

Offset: 1

Gus Wiseman, Jan 02 2019

nonn

Except for n = 2, same as A182850. Unlike A182850, the terms of this sequence depend only on the prime signature (A101296, A118914) of the index.

Antti Karttunen, Table of n, a(n) for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Positions of 1's are the prime numbers A000040.
Positions of 2's are the proper prime powers A246547.
Positions of 3's are A182853.
Row lengths of A323023.
Cf. A001221, A001222, A046523, A056239, A070175, A071625, A101296, A112798, A118914, A181819, A181821, A182850, A182857, A304465, A323022.

dep[n_]:=If[n==1,0,If[PrimeQ[n],1,1+dep[Times@@Prime/@Last/@FactorInteger[n]]]];
Array[dep,100]

A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A323014(n) = if(1==n,0,if(isprime(n),1, 1+A323014(A181819(n)))); \\ Antti Karttunen, Jun 10 2022

For all n >= 1, a(n) = a(A046523(n)). [See comment] - Antti Karttunen, Jun 10 2022

Terms a(88) and beyond from Antti Karttunen, Jun 10 2022

A304818 If n = Product_i p(y_i) where p(i) is the i-th prime number and y_i <= y_j for i < j, then a(n) = Sum_i y_i*i.

Original entry on oeis.org

0, 1, 2, 3, 3, 5, 4, 6, 6, 7, 5, 9, 6, 9, 8, 10, 7, 11, 8, 12, 10, 11, 9, 14, 9, 13, 12, 15, 10, 14, 11, 15, 12, 15, 11, 17, 12, 17, 14, 18, 13, 17, 14, 18, 15, 19, 15, 20, 12, 16, 16, 21, 16, 19, 13, 22, 18, 21, 17, 21, 18, 23, 18, 21, 15, 20, 19, 24, 20, 19

Offset: 1

Gus Wiseman, May 18 2018

If n > 1 is not a prime number, we have a(n) >= A056239(n) >= Omega(n) >= omega(n) >= A071625(n) >= ... >= Omicron(n) >= omicron(n) > 1, where Omega = A001222, omega = A001221, Omicron = A304687 and omicron = A304465.

			The multiset of prime indices (see A112798) of 216 is {1,1,1,2,2,2}, which becomes {1,2,3,4,4,5,5,6,6} under A304660, so a(216) = 1+2+3+4+4+5+5+6+6 = 36.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000720, A001221, A001222, A001358, A055932, A056239, A071625, A112798, A181819, A182850, A182857, A275870, A304465, A304660.

a:= n-> (l-> add(i*numtheory[pi](l[i]), i=1..nops(l)))(
             sort(map(i-> i[1]$i[2], ifactors(n)[2]))):
seq(a(n), n=1..100);  # Alois P. Heinz, May 20 2018

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[With[{y=primeMS[n]},Sum[y[[i]]*i,{i,Length[y]}]],{n,20}]

a(n) = {my(f = factor(n), s = 0, i = 0); for (k=1, #f~, for (kk = 1, f[k, 2], i++; s += i*primepi(f[k,1]););); s;} \\ Michel Marcus, May 19 2018

vf(n) = {my(f=factor(n), nb = bigomega(n), g = vector(nb), i = 0); for (k=1, #f~, for (kk = 1, f[k, 2], i++; g[i] = primepi(f[k,1]););); return(g);} \\ A112798
a(n) = {my(g = vf(n)); sum(k=1, #g, k*g[k]);} \\ Michel Marcus, May 19 2018

a(n) = A056239(A304660(n)).

A323023 Irregular triangle read by rows where row n is the omega-sequence of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 1, 3, 1, 2, 1, 2, 2, 1, 1, 3, 2, 2, 1, 1, 2, 2, 1, 2, 2, 1, 4, 1, 1, 3, 2, 2, 1, 1, 3, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 2, 2, 1, 3, 1, 3, 2, 2, 1, 1, 3, 3, 1, 1, 5, 1, 2, 2, 1, 2, 2, 1, 2, 2, 1, 4, 2, 1, 1, 2, 2, 1, 2, 2

Offset: 1

Gus Wiseman, Jan 02 2019

We define the omega-sequence of n to have length A323014(n), and the k-th term is Omega(red^{k-1}(n)), where Omega = A001222 and red^{k} is the k-th functional iteration of A181819.

Except for n = 1, all rows end with 1. If n is not prime, the term in row n prior to the last is A304465(n).

			The sequence of omega-sequences begins:
   1:            26: 2 2 1      51: 2 2 1        76: 3 2 2 1
   2: 1          27: 3 1        52: 3 2 2 1      77: 2 2 1
   3: 1          28: 3 2 2 1    53: 1            78: 3 3 1
   4: 2 1        29: 1          54: 4 2 2 1      79: 1
   5: 1          30: 3 3 1      55: 2 2 1        80: 5 2 2 1
   6: 2 2 1      31: 1          56: 4 2 2 1      81: 4 1
   7: 1          32: 5 1        57: 2 2 1        82: 2 2 1
   8: 3 1        33: 2 2 1      58: 2 2 1        83: 1
   9: 2 1        34: 2 2 1      59: 1            84: 4 3 2 2 1
  10: 2 2 1      35: 2 2 1      60: 4 3 2 2 1    85: 2 2 1
  11: 1          36: 4 2 1      61: 1            86: 2 2 1
  12: 3 2 2 1    37: 1          62: 2 2 1        87: 2 2 1
  13: 1          38: 2 2 1      63: 3 2 2 1      88: 4 2 2 1
  14: 2 2 1      39: 2 2 1      64: 6 1          89: 1
  15: 2 2 1      40: 4 2 2 1    65: 2 2 1        90: 4 3 2 2 1
  16: 4 1        41: 1          66: 3 3 1        91: 2 2 1
  17: 1          42: 3 3 1      67: 1            92: 3 2 2 1
  18: 3 2 2 1    43: 1          68: 3 2 2 1      93: 2 2 1
  19: 1          44: 3 2 2 1    69: 2 2 1        94: 2 2 1
  20: 3 2 2 1    45: 3 2 2 1    70: 3 3 1        95: 2 2 1
  21: 2 2 1      46: 2 2 1      71: 1            96: 6 2 2 1
  22: 2 2 1      47: 1          72: 5 2 2 1      97: 1
  23: 1          48: 5 2 2 1    73: 1            98: 3 2 2 1
  24: 4 2 2 1    49: 2 1        74: 2 2 1        99: 3 2 2 1
  25: 2 1        50: 3 2 2 1    75: 3 2 2 1     100: 4 2 1

Row lengths are A323014, or A182850 if we assume A182850(2) = 1.
First column is empty if n = 1 and otherwise A001222(n).
Second column is empty if n is 1 or prime and otherwise A001221(n).
Third column is empty if n is 1, prime, or a power of a prime and otherwise A071625(n).
Cf. A024619, A056239, A067340, A118914, A181819, A181821, A182857, A304464, A304465, A323022.

red[n_]:=Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]];
omg[n_,k_]:=If[k==1,PrimeOmega[n],omg[red[n],k-1]];
dep[n_]:=If[n==1,0,If[PrimeQ[n],1,1+dep[Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]]]]];
Table[omg[n,k],{n,100},{k,dep[n]}]

A182857 Smallest number that requires exactly n iterations to reach a fixed point under the x -> A181819(x) map.

Original entry on oeis.org

1, 3, 4, 6, 12, 60, 2520, 1286485200, 35933692027611398678865941374040400000

Offset: 0

Matthew Vandermast, Jan 05 2011

nonn

a(9) has 296 digits.
Related to Levine's sequence (A011784): A011784(n) = A001222(a(n)) = A001221(a(n+1)) = A051903(a(n+2)) = A071625(a(n+2)). Also see A182858.
Values of n where A182850(n) increases to a record.
The multiplicity of prime(k) in a(n+1) is the k-th largest prime index of a(n), which is A296150(a(n),k). - Gus Wiseman, May 13 2018

			From _Gus Wiseman_, May 13 2018: (Start)
Like A001462 the following sequence of multisets whose Heinz numbers belong to this sequence is a run-length describing sequence, as the number of k's in row n + 1 is equal to the k-th term of row n.
{2}
{1,1}
{1,2}
{1,1,2}
{1,1,2,3}
{1,1,1,2,2,3,4}
{1,1,1,1,2,2,2,3,3,4,4,5,6,7}
{1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7,7,7,8,8,9,9,10,10,11,12,13,14}
(End)

Gus Wiseman, Table of n, a(n) for n = 0..9

Cf. A001462, A007755, A007916, A009287, A012257, A112798, A181819, A182850-A182858, A296150, A304455, A304464, A304465.

Prepend[Function[m,Times@@Prime/@m]/@NestList[Join@@Table[Table[i,{Reverse[#][[i]]}],{i,Length[#]}]&,{2},8],1] (* Gus Wiseman, May 13 2018 *)

For n > 0, a(n) = A181819(a(n+1)). For n > 1, a(n) = A181821(a(n-1)).

A323022 Fourth omega of n. Number of distinct multiplicities in the prime signature of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, Jan 02 2019

nonn

The indices of terms greater than 1 are {60, 84, 90, 120, 126, 132, 140, 150, ...}.
First term greater than 2 is a(1801800) = 3. In general, the first appearance of k is a(A182856(k)) = k.
The prime signature of n (row n of A118914) is the multiset of prime multiplicities in n.
We define the k-th omega of n to be Omega(red^{k-1}(n)) where Omega = A001222 and red^{k} is the k-th functional iteration of A181819. The first three omegas are A001222, A001221, A071625, and this sequence is the fourth. The zeroth omega is not uniquely determined from prime signature, but one possible choice is A056239 (sum of prime indices).

			The prime signature of 1286485200 is {1, 1, 1, 2, 2, 3, 4}, in which 1 appears three times, two appears twice, and 3 and 4 both appear once, so there are 3 distinct multiplicities {1, 2, 3} and hence a(1286485200) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A001221, A001222, A006939, A025487, A056239, A059404, A062770, A071625, A118914, A181819, A182856, A182857, A304464, A304465, A323014, A323023.

red[n_]:=Times@@Prime/@Last/@If[n==1,{},FactorInteger[n]];
Table[PrimeNu[red[red[n]]],{n,200}]

a(n) = my(e=factor(n)[, 2], s = Set(e), m=Map(), v=vector(#s)); for(i=1, #s, mapput(m,s[i],i)); for(i=1, #e, v[mapget(m,e[i])]++); #Set(v) \\ David A. Corneth, Jan 02 2019

A071625(n) = #Set(factor(n)[, 2]); \\ From A071625
A181819(n) = factorback(apply(e->prime(e),(factor(n)[,2])));
A323022(n) = A071625(A181819(n)); \\ Antti Karttunen, Jan 03 2019

a(n) = A071625(A181819(n))
     = A001221(A181819(A181819(n)))
     = A001222(A181819(A181819(A181819(n))))
     = A056239(A181819(A181819(A181819(A181819(n))))).

More terms from Antti Karttunen, Jan 03 2019

A325272 Adjusted frequency depth of n!.

Original entry on oeis.org

0, 1, 3, 4, 5, 4, 6, 6, 6, 4, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 6, 7, 7, 7, 7, 7, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 6, 6, 6, 6, 7, 7, 7, 6, 6, 6, 6, 7, 7, 7, 8, 7, 7, 7, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 1

Gus Wiseman, Apr 18 2019

nonn

The adjusted frequency depth of a positive integer n is 0 if n = 1, and otherwise it is one plus the number of times one must apply A181819 to reach a prime number, where A181819(k = p^i*...*q^j) = prime(i)*...*prime(j) = product of primes indexed by the prime exponents of k. For example, 180 has adjusted frequency depth 5 because we have: 180 -> 18 -> 6 -> 4 -> 3.

			Recursively applying A181819 starting with 120 gives 120 -> 20 -> 6 -> 4 -> 3, so a(5) = 5.

a(n) = A001222(A325275(n)).
Cf. A000142, A006939, A303555, A323023, A325238, A325273, A325274, A325276, A325277.
Omega-sequence statistics: A001222 (first omega), A001221 (second omega), A071625 (third omega), A323022 (fourth omega), A304465 (second-to-last omega), A182850 or A323014 (length/frequency depth), A325248 (Heinz number), A325249 (sum).

fd[n_]:=Switch[n,1,0,?PrimeQ,1,,1+fd[Times@@Prime/@Last/@FactorInteger[n]]];
Table[fd[n!],{n,30}]

a(n) = A323014(n!).

cp's OEIS Frontend

A304634 Numbers n with prime omicron 2, meaning A304465(n) = 2.

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A304636 Numbers n with prime omicron 3, meaning A304465(n) = 3.

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A181819 Prime shadow of n: a(1) = 1; for n>1, if n = Product prime(i)^e(i), then a(n) = Product prime(e(i)).

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A182850 a(n) = number of iterations that n requires to reach a fixed point under the x -> A181819(x) map.

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A323014 a(1) = 0; a(prime) = 1; otherwise a(n) = 1 + a(A181819(n)).

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A304818 If n = Product_i p(y_i) where p(i) is the i-th prime number and y_i <= y_j for i < j, then a(n) = Sum_i y_i*i.

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A323023 Irregular triangle read by rows where row n is the omega-sequence of n.

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